Appendix A Some Concepts from Algebra This appendix contains precise statements of various algebraic facts and definitions used in the text. For students who have had a course in abstract algebra, much of this material will be familiar. For students seeing these terms for the first time, keep in mind that the abstract concepts defined here are used in the text in very concrete situations. §1 Fields and Rings We first give a precise definition of a field. Definition 1. A field consists of a set k and two binary operations “+” and “·” de- fined on k for which the following conditions are satisfied: (i) (a+b)+c = a+(b+c) and (a·b)·c = a·(b·c) for all a, b, c ∈ k (associativity). (ii) a + b = b + a and a · b = b · a for all a, b ∈ k (commutativity). (iii) a · (b + c)=a · b + a · c for all a, b, c ∈ k (distributivity). (iv) There are 0, 1 ∈ k such that a + 0 = a · 1 = a for all a ∈ k (identities). (v) Given a ∈ k, there is b ∈ k such that a + b = 0 (additive inverses). (vi) Given a ∈ k, a = 0, there is c ∈ k such that a · c = 1 (multiplicative inverses). The fields most commonly used in the text are Q, R, and C. In the exercises to §1 of Chapter 1, we mention the field F2 which consists of the two elements 0 and 1. Some more complicated fields are discussed in the text. For example, in §3 of Chapter 1, we define the field k(t1,...,tm) of rational functions in t1,...,tm with coefficients in k. Also, in §5 of Chapter 5, we introduce the field k(V) of rational functions on an irreducible variety V. If we do not require multiplicative inverses, then we get a commutative ring. Definition 2. A commutative ring consists of a set R and two binary operations “+” and “·” defined on R for which the following conditions are satisfied: © Springer International Publishing Switzerland 2015 593 D.A. Cox et al., Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, DOI 10.1007/978-3-319-16721-3 594 Appendix A Some Concepts from Algebra (i) (a + b)+c = a +(b + c) and (a · b) · c = a · (b · c) for all a, b, c ∈ R (associativity). (ii) a + b = b + a and a · b = b · a for all a, b ∈ R (commutativity). (iii) a · (b + c)=a · b + a · c for all a, b, c ∈ R (distributivity). (iv) There are 0, 1 ∈ R such that a + 0 = a · 1 = a for all a ∈ R (identities). (v) Given a ∈ R, there is b ∈ R such that a + b = 0 (additive inverses). Note that any field is obviously a commutative ring. Other examples of commu- tative rings are the integers Z and the polynomial ring k[x1,...,xn]. The latter is the most commonly used ring in the book. In Chapter 5, we construct two other com- mutative rings: the coordinate ring k[V] of polynomial functions on an affine variety V and the quotient ring k[x1,...,xn]/I, where I is an ideal of k[x1,...,xn]. A special case of commutative rings are the integral domains. Definition 3. A commutative ring R is an integral domain if whenever a, b ∈ R and a · b = 0, then either a = 0orb = 0. A zero divisor in a commutative ring R is a nonzero element a ∈ R such that a · b = 0 for some nonzero b ∈ R. Hence integral domains have no zero divisors. Any field is an integral domain, and the polynomial ring k[x1,...,xn] is an integral domain. In Chapter 5, we prove that the coordinate ring k[V] of a variety V is an integral domain if and only if V is irreducible. Finally, we note that the concept of ideal can be defined for any ring. Definition 4. Let R be a commutative ring. A subset I ⊆ R is an ideal if it satisfies: (i) 0 ∈ I. (ii) If a, b ∈ I, then a + b ∈ I. (iii) If a ∈ I and b ∈ R, then b · a ∈ I. Note how this generalizes the definition of ideal given in §4 of Chapter 1. §2 Unique Factorization Definition 1. Let k be a field. A polynomial f ∈ k[x1,...,xn] is irreducible over k if f is nonconstant and is not the product of two nonconstant polynomials in k[x1,...,xn]. This definition says that if a nonconstant polynomial f is irreducible over k, then up to a constant multiple, its only nonconstant factor is f itself. Also note that the concept of irreducibility depends on the field. For example, x2 +1 is irreducible over Q and R, but over C we have x2 + 1 =(x − i)(x + i). Every nonconstant polynomial is a product of irreducible polynomials as follows. Theorem 2. Every nonconstant f ∈ k[x1,...,xn] can be written as a product f = f1 ·f2 ···fr of irreducibles over k. Further, if f = g1 ·g2 ···gs is another factorization into irreducibles over k, then r = s and the gi’s can be permuted so that each fi is a nonzero constant multiple of gi. §3 Groups 595 The final assertion of the theorem says that unique factorization holds in the polynomial ring k[x1,...,xn]. Proof. The proof of Theorem 2 is by induction on the number of variables. The base case k[x1] is covered in §5 of Chapter 1. Now suppose that k[x1,...,xn−1] has unique factorization. The key tool for proving unique factorization in k[x1,...,xn] is Gauss’s Lemma, which in our situation can be stated as follows. Proposition 3. Let k(x1,...,xn−1) be the field of rational functions in x1,...,xn−1. If f ∈ k[x1,...,xn] is irreducible and has positive degree in xn, then f is irreducible in k(x1,...,xn−1)[xn]. This follows from Proposition 5 of Section 9.3 of DUMMIT and FOOTE (2004) since k[x1,...,xn−1] has unique factorization. Combining Proposition 3 with unique factorization in the rings k[x1,...,xn−1] and k(x1,...,xn−1)[xn], it is straightforward to prove that k[x1,...,xn] has unique factorization. See Theorem 7 of Section 9.3 of DUMMIT and FOOTE (2004)forthe details. For polynomials in Q[x1,...,xn], there are algorithms for factoring into irre- ducibles over Q. A classical algorithm due to Kronecker is discussed in Theorem 4.8 of MINES,RICHMAN, and RUITENBERG (1988), and a more efficient method is given in Section 16.6 of VON ZUR GATHEN and GERHARD (2013). Most computer algebra systems have a command for factoring polynomials in Q[x1,...,xn]. Factoring polynomials in R[x1,...,xn] or C[x1,...,xn] is much more difficult. §3 Groups A group can be defined as follows. Definition 1. A group consists of a set G and a binary operation “·” defined on G for which the following conditions are satisfied: (i) (a · b) · c = a · (b · c) for all a, b, c ∈ G (associativity). (ii) There is 1 ∈ G such that 1· a = a · 1 = a for all a ∈ G (identity). (iii) Given a ∈ G, there is b ∈ G such that a · b = b · a = 1 (inverses). A simple example of a group is given by the integers Z under addition. Note Z is not a group under multiplication. A more interesting example comes from linear algebra. Let k be a field and define GL(n, k)={A | A is an invertible n × n matrix with entries in k}. From linear algebra, we know that the product AB of two invertible matrices A and B is again invertible. Thus, matrix multiplication defines a binary operation on GL(n, k), and it is easy to verify that all of the group axioms are satisfied. In Chapter 7, we will need the notion of a subgroup. 596 Appendix A Some Concepts from Algebra Definition 2. Let G be a group. A nonempty subset H ⊆ G is called a subgroup if it satisfies: (i) 1 ∈ H. (ii) If a, b ∈ H, then a · b ∈ H. (iii) If a ∈ H, then a−1 ∈ H, where a−1 is the inverse of a in G. One important group is the symmetric group Sn.Letn be a positive integer and consider the set Sn = {σ : {1,..., n}→{1,...,n}|σ is one-to-one and onto}. Then composition of functions turns Sn into a group. Since an element σ ∈ Sn permutes the numbers 1 through n, we call σ a permutations. Note that Sn has n! elements. A transposition is an element of Sn that interchanges two numbers in {1,...,n} and leaves all other numbers unchanged. Every permutation is a product of transpo- sitions, though not in a unique way. The sign of a permutation is defined to be +1ifσ is a product of an even number of transpositions, sgn(σ)= −1ifσ is a product of an odd number of transpositions. One can show that sgn(σ) is well-defined. Proofs of these assertions about Sn can be found in Section 3.5 of DUMMIT and FOOTE (2004). §4 Determinants In linear algebra, one usually encounters the determinant det(A) of an n × n matrix A with entries in a field such as R or C. Typical formulas are a11 a12 det = a11a22 − a12a21 a21 a22 and ⎛ ⎞ a11 a12 a13 ⎝ ⎠ a22 a23 a21 a23 a21 a22 det a21 a22 a23 = a11 det − a12 det + a13 det , a32 a33 a31 a33 a31 a32 a31 a32 a33 which simplifies to a11a22a33 − a11a23a32 − a12a21a33 + a12a23a31 + a13a21a32 + a13a22a31.